In the simulations, we use random initial value data with \(100 \times 100\)  mesh so that the size of the input data is \(102 \times 102\)  containing a pad as a boundary condition. Also, \(N=3\)  (Heat, Fisher's, AC) or 9 (Sine, Tanh) for \(\epsilon (\phi ^n)\)  is fixed depending on given equations and a \(3 \times 3\)  convolutional filter is used with the stride of 1 in Eq. (REF ). Hence, the filter has 10,000 (\(=((100+2-3)/1+1)^2\) ) chances to learn the evolution of results images, so training a model using only two consecutive images are enough to optimize nine or thirteen model parameters (\(w_0,\cdots , w_4, a_0,\cdots ,a_3\) ). As an optimizer, ADAM [1]} is used with a learning rate of 0.01 and without any regularization. Instead, we apply early stopping [2]} based on a validation data to avoid overfitting. To demonstrate the approximation \(\epsilon (\phi ^n)\)  for non-polynomial functions \(f(\phi ^n)\) , we additionally consider sine and tanh functions besides heat, Fisher's, and AC equations.
